6 MULTILINEAR MAPPINGS. DETERMINANTS 375
(A.6.7) det(w) ^ 0 if and only ifu is bijective.
If u is bijective, it has an inverse u~~l such that u 0 w"1 = 1E. Hence, by
(A.6.6) and (A.6.5.2), we have det(w) det^/"1) = 1, so that certainly det(w) ^ 0.
If w is not bijective, then it is not injective (A.4.19), hence there exists bl ^ 0
such that u(b^ = 0. There exists a basis (bi\^i^n of E containing bt (AA5),
and we have fQ(bls..., bn) ^0, whereas/0(w(^),..., w(6J) = 0. Hence
det(w) = 0.
(A.6.8) Let (bi\ «- £<w be a basis of E, and let M(u) = (o^-) be the matrix of w
with respect to the two bases (bt) and (bt) of E (or, as is usually said, the matrix
of u with respect to the basis (&,-)). Since /0(^i, ..., bn) ^ 0, the formulas
(A.6.1.1)and (A.6.5.1) give
(A.6.8.1) det(w) = ^ £ff otff(1)iaa(2)2 • - • aff(n)n,
a
where a runs through the symmetric group Sw of all permutations of
{1,2,...,*}.
The determinant of the matrix M(u) is by definition the determinant of u.
This provides the link between our theory and the classical theory of deter-
minants in its original form. We shall not need to use this latter theory, and
we leave the task of transcribing our results into the old-fashioned notation
to those readers who are interested in this type of calculation. In applications
it is always much simpler to go back to the definition (A.6.5), as we shall
illustrate by considering the eigenvalues of an endomorphism.
(A.6.9) The definition of the eigenvalues of an endomorphism u is that given
in (11.1.1), except that the field C is now replaced by an arbitrary field K.
It follows immediately from (A.6.7) that these eigenvalues are the roots of the
equation (called the characteristic equation of u)
The formula (A.6.8.1) shows immediately that the left-hand side of this
equation is a polynomial of degree n in A, with leading coefficient (— l)n.
In what follows we shall assume that the field K is algebraically closed, so
that det(w — A • 1E) factorizes into linear factors (/^ — /L)(A2 — X) • • * (/lrt — A).
(A.6.10)* There exists a basis (biy ..., bn) ofE such that